commutative algebra

The function is an example of a cubic function with leading coefficient −7 and constant coefficient 3. ...

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In commutative algebra, one major focus of study is divisibility among polynomials. If R is an integral domain and f and g are polynomials in R[X], it is said that f divides g if there exists a polynomial q in R[X] such that f q = g. One can then show that "every zero gives rise to a linear factor", or more formally: if f is a polynomial in R[X] and r is an element of R such that f(r) = 0, then the polynomial (X − r) divides f. The converse is also true. The quotie ...

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One also speaks of polynomials in several variables, obtained by taking the ring of polynomials of a ring of polynomials: R[X,Y] = (R[X])[Y] = (R[Y])[X]. These are of fundamental importance in algebraic geometry which studies the simultaneous zero sets of several such multivariate polynomials. Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest ...

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There are many open questions about prime numbers. The most significant of these is the Riemann hypothesis, which essentially says that the primes are as regularly distributed as possible. From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about 1/ log x of number less than x are primes, the prime number theorem) also holds for much shorter intervals of length a ...

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The Sieve of Eratosthenes is a simple way and the Sieve of Atkin a fast way to compute the list of all prime numbers up to a given limit. In practice though, one usually wants to check if a given number is prime, rather than generate a list of primes. Further, it is often satisfactory to know the answer with a high probability. It is possible to quickly check whether a given large number (say, up to a few thousand digits) is prime using probabilistic primality tests. These typically pick a random number called a "witness" and check so ...

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There are infinitely many prime numbers. The oldest known proof for this statement is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20). Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following: Suppose you have a finite number of primes. Call this number m. Multiply all m primes together and add one (see Euclid number). The resulting number is not divisible by any of the finite set of primes, because dividing by any o ...

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is a horizontal line with y-intercept a0. The graph of a degree 1 polynomial function (or linear function) f(x) = a0 + a1x, where is an oblique line with y-intercept a0 and slope See also:

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For given constants (i.e., numbers) a0, …, an in some field (possibly but not limited to R or C) with an non-zero, for n > 0, then a polynomial (function) of degree n is a function of the form More concisely, a polynomial function can be written in sigma notation as The constants a0, …, an are called the co ...

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Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or R-submodule, to be more explicit) if, for any n in N and any r in R, the product rn is in N (or nr for a right module). If M and N are left R-modules, then a map f : M → N is a homomorphism of R-modules if, for any m, n in M and r, s in R, f(rmSee also:

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Specifically, a left module over the ring R consists of an abelian group (M, +) and an operation R × M → M (called scalar multiplication, usually just written by juxtaposition, i.e. as rx for r in R and x in M) such that For all r,s in R, x,y in M, we have r(x+y) = rx+ry (r+s)x = rx+sx (rs)x = rSee also:

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In algebraic geometry and commutative algebra, a singularity is a prime ideal whose localization is not a regular local ring (alternately a scheme (mathematics) with a stalk that is not a regular local ring). For example, y2 − x3 = 0 defines an isolated singular point (at the cusp) x = y = 0. The ring in question is given by The maximal ideal of the localization at (t2,t3See also:

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Finitely generated. A module M is finitely generated if there exist finitely many elements x1,...,xn in M such that every element of M is a linear combination of those elements with coefficients from the scalar ring R. Free. A free module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the scalar ring R. These are the modules that behave very much like vector spaces. Projective. Projective modules are direct summands of fre ...

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If M is a left R-module, then the action of an element r in R is defined to be the map M → M that sends each x to rx (or xr in the case of a right module), and is necessarily a group endomorphism of the abelian group (M,+). The set of all group endomorphisms of M is denoted EndZ(M) and forms a ring under addition and composition, and sending a ring element r of R to its action actually define ...

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Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. Some polynomials, such as f(x) = x² + 1, do not have any roots among the real numbers. If, however, the set of allowed candidates is expanded to the complex numbers, every (non-constant) polynomial has a root: this is the statement of the fundamental theorem of algebra. There is a difference between approximating roots and finding concrete closed formulas for them. Formulas for the roots of polynom ...

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Also in linear algebra, the characteristic polynomial of a square matrix encodes several important properties of the matrix. In graph theory the chromatic polynomial of a graph encodes the different ways to vertex color the graph using x colors. In abstract algebra, one may define polynomials with coefficients in any ring. In knot theory the Alexander polynomial, the Jones ...

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Any ring R can be viewed as a preadditive category with a single object. With this understanding, a left R-module is nothing but a (covariant) additive functor from R to the category Ab of abelian groups. Right R-modules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C; these functors form a functor category C-Mod which is the natural gene ...

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The largest known prime, as of December 2005, is 230402457 − 1 (this number is 9,152,052 digits long); it is the 43rd known Mersenne prime. M30402457 was found on December 15, 2005 by Curtis Cooper and Steven Boone, professors at Central Missouri State University and members of a collaborative effort known as GIMPS. The next largest known prime is 225964951 − 1 (this number is 7,816,230 digits long); it is the 42nd known Mersenne prime. M25964951 was found on Feb ...

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